A portion of this paper was originally published at the I\({}^2\) MTC conference in 2021 [1]}. There, the authors showed the advantages of applying the method of variable projection [2]}, [3]} to identifying the four-parameter model for a sine wave, see the IEEE-standard 1057 [4]} for an exact definition of the model. This model is commonly used in waveform digitizers, e.g. digital oscilloscopes, analog to digital converters etc [5]}, [6]}. However,
the authors previous work was primarily motivated by the widespread availability of low-cost MEMS accelerometers and gyroscopes [7]}; which, commonly create such signals. The ubiquitous applicability of these sensors  [8]}, [9]}, [10]} calls forth the need for stable and computationally efficient methods.
